Proper holomorphic maps from the unit disk to some unit ball

TL;DR

This paper investigates proper rational maps from the unit disk to higher-dimensional balls, establishing a normal form for their equivalence classes and proving homotopy results in higher dimensions.

Contribution

It introduces a normal form for polynomial proper maps and proves that all rational proper maps are homotopic in dimensions two or higher.

Findings

Normal form for polynomial proper maps established

All rational proper maps are homotopic in target dimension ≥ 2

Analysis of the moduli space of unitary equivalence classes

Abstract

We study proper rational maps from the unit disk to balls in higher dimensions. After gathering some known results, we study the moduli space of unitary equivalence classes of polynomial proper maps from the disk to a ball, and we establish a normal form for these equivalence classes. We also prove that all rational proper maps from the disk to a ball are homotopic in target dimension at least $2$.